Genesis 1:24-27 states that humans were created after the other animals, and the first man and woman were created simultaneously.

1:24 And God said, Let the earth bring forth the living creature after his kind, cattle, and creeping thing, and beast of the earth after his kind: and it was so.1:25 And God made the beast of the earth after his kind, and cattle after their kind, and every thing that creepeth upon the earth after his kind: and God saw that it was good.1:26 And God said, Let us make man in our image, after our likeness: and let them have dominion over the fish of the sea, and over the fowl of the air, and over the cattle, and over all the earth, and over every creeping thing that creepeth upon the earth.1:27 So God created man in his own image, in the image of God created he him; male and female created he them.

While Genesis 2:18-22 states that man was created first, then the animals and only then the woman from the man's rib.

2:18 And the LORD God said, It is not good that the man should be alone; I will make him an help meet for him.2:19 And out of the ground the LORD God formed every beast of the field, and every fowl of the air; and brought them unto Adam to see what he would call them: and whatsoever Adam called every living creature, that was the name thereof.2:20 And Adam gave names to all cattle, and to the fowl of the air, and to every beast of the field; but for Adam there was not found an help meet for him.2:21 And the LORD God caused a deep sleep to fall upon Adam, and he slept: and he took one of his ribs, and closed up the flesh instead thereof;2:22 And the rib, which the LORD God had taken from man, made he a woman, and brought her unto the man.

Assuming that only one story is "correct" is very problematic, and by asking in this way the question is just soliciting opinions.
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curiousdanniiJan 14 at 22:15

1:27 identifies man and woman as God's last creation. 2:18 is giving details. At this point Adam, but not Eve, has been created. God presents the animals to Adam for naming, and to demonstrate to him that none of them are similar enough to him to be a companion. It is mentioned that God created them, an event which had already occurred at that time. This is only an opinion, not an answer. For a real answer, you need to modify the question so that it can be answered contextually from the Bible (possibly on the Hermeneutics site) or according to the teaching of specific Christian groups.
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discipleMar 11 at 3:04

3 Answers
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In verse 19, it also talks about how formed the fowls of the air, while in chapter one, it says that they were created on day 5, a day before man. Therefore, it can be inferred that that there is shift between verses 18 and 19, while it goes back in time a little to give a better understanding of how he noticed that he needed one. While the first chapter was about detailing what happened in what order over the course of days, chapter two is much more detailed, but not entirely chronological. Also, since Eve was made from a rib taken from Adam after he noticed his lack of a female counterpart, she was obviously made after.

In order to understand the meaning of any text, you must understand it's purpose. If you don't understand why a bit of text was written, you are quite likely to come away with a bad understanding of what it means.

In this case your question assumes a mis-understanding of the purpose of the two texts in question. If you correct that understanding, it becomes apparent that there is actually no contradiction and nothing that needs resolving from an inerrancy standpoint.

The two chapters you quote from in Genesis serve slightly different purposes. It is quite common on ancient Hebrew literature to repeat parts of a story in different ways to draw out different points. In this case one passage tells us something about the Chronology of events, the second goes on to give us another event that happens to be sandwiched inbetween the last steps of the first event that includes some additional commentary.

If you want to know in what order things were created, Genesis 1 should be your guide.

In Genesis 2, we find out that that (presumably about day 6) God was trying to show Adam that his own personal help-mate was going to be a different kind of creature than the animals, so after he makes Adam but before he makes Eve he parades the rest of the created order before Adam and demonstrates that none of them are like he is. The animals are not made on the spot, there is just a bit of commentary that reminds us that they were created out of the ground. In the process, all the animals get names.

But then we go on to the creation of Eve, made not out of the ground but out of Adam's flesh. This second chapter helps us establish some order or rank to creation as well as lay the foundation for its purpose, not re-sequence the events of creation.

The creation story in Genesis 1:1-2:4a is written in a different style than the more primitive creation story in Genesis 2:4b-25. The first creation story is consistent in style and language with other passages attributed to the Priestly Source, and is therefore also attributed to the same source. The second creation story is consistent in style and language with other passages attributed to the anonymous source now known as the Yahwist. Although the documentary hypothesis is no longer considered as simple and straight-forward an explanation of the development of the Pentateuch as it was once believed, in one form or another the hypothesis remains the best explanation we have.

Leon R. Kass says in The Beginning of Wisdom: Reading Genesis, at page 56, that once we recognise the independence of the two creation stories, we are compelled to adopt a critical principle of reading if we mean to understand each story on its own terms. We must scrupulously avoid reading into the second story any facts or notions taken from the first, and vice versa.

In the first story, animals are created before man and in the second story, the man is created before the animals. As Kass has inferred, this should not be seen as a contradiction, but the inevitable result of two different stories being written by two different authors. There is no need to regard either one as more correct than the other.